Simplify the following expression and state the condition under which the simplification is valid: $z = \dfrac{x^2 - 5x + 6}{x^2 + 3x - 10}$
Answer: First factor the expressions in the numerator and denominator. $ \dfrac{x^2 - 5x + 6}{x^2 + 3x - 10} = \dfrac{(x - 3)(x - 2)}{(x + 5)(x - 2)} $ Notice that the term $(x - 2)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(x - 2)$ gives: $z = \dfrac{x - 3}{x + 5}$ Since we divided by $(x - 2)$, $x \neq 2$. $z = \dfrac{x - 3}{x + 5}; \space x \neq 2$